Function properties

• A set is a collection of objects, called its elements.

• A class is a membership-restricted special set.

• A type is a class of objects, called values.

• A relation is a special set (class).

A function is a

• set

• set of ordered pairs

• relation

relation is a set type is a set relation is a set of ordered pairs

function is a set function is a relation function is a set of ordered pairs function is a type (function type)

Properties of functions

A function is a special set. A function is a special type of relation. A type is a set. A function is a mapping between two sets (types). A relation is a set of ordered pairs (a,b).

A function is defined in terms of sets: a function between two sets A and B (or a function on a set A, in case A = B), is a special type of relation between A and B;

Any binary relation between two sets A and B is a set, 𝓡 = { (a,b) | a ∈ A ∧ b ∈ B }, of ordered pairs, (a,b) such that the first component, a, is from the domain and the second, b, is from the codomain.

a𝓡b is the notation meaning that a is 𝓡-related to b; another notation, (a,b) ∈ 𝓡, means the same.

A function f : ℝ → ℝ with ∀x,y ∈ ℝ is

• order-preserving : (x <= y) -> (f x <= f y)

• metric-preserving : |x − y| -> |f x − f y|

• addition-preserving: f (x + y) -> f x + f y

A function f : ℝ → ℝ with ∀x,y ∈ ℝ (is it ℝ or any set?) is monotonic if the elements x and y are covariant and the results f x and f y stay covariant. For example, (x <= y) -> (f x <= f y). A function is non-monotonic is the relation is reversed or contravariant, e.g. (x <= y) -> (f x > f y). Some functions are neither or both; e.g. f : ℝ → ℝ, f x = x² is neither; it is non-monotonic for x ∈ (-1,0) ∨ x ∈ (0,1) and monotonic for other values.

x	x²
-2.0	4
-1.5	2.25
-0.1	0.01
-0.01	0.0001
0	0
0.01	0.0001
0.1	0.01
0.4	0.16
0.5	0.25
2	4